Question

A bead slides down along a piece of wire that is in the shape of a helix. The helix lies on the surface of a vertical cylinder of radius r, and the vertical distance between turns is d. (a) Ordinarily when an object slides downhill under the influence of kinetic friction, the velocity-independence of kinetic friction implies that the acceleration is constant, and therefore there is no limit to the object’s velocity. Explain the physical reason why this argument fails here, so that the bead will in fact have some limiting velocity. (b) Find the limiting velocity. (c) Show that your result has the correct behavior in the limit of r → ∞.

Answer 1

given piece of wire is like a helix
hence
radius of helix = R
distance between sucessive loops = h

now, at some point t, when speed of beat is v
centripital acceleration = v^2/R
tangential acceleration = dv/dt

hence tangential friction = f
centripital force friciton = F
then from force balance
F = uN = umv^2/R
f = mg*sin(theta) - mdv/dt - umv^2/R
but f = u*mg*cos(theta)
hence

dv/dt = (u/R)(Rg*sin(theta)/u - Rg*cos(theta)) - v^2)

integrating
((v + sqrt(Rg*sin(theta)/u - Rg*cos(theta)))/(v - sqrt(Rg*sin(thedta)/u - Rg*cos(theta)))) = e^(2*sqrt(Rg*sin(theta)/u - Rg*cos(theta))*ut/R)
v = ( e^(2*sqrt(Rg*sin(theta)/u - Rg*cos(theta))*ut/R) + 1)/( e^(2*sqrt(Rg*sin(theta)/u - Rg*cos(theta))*ut/R) - 1)
v = sqrt(Rg*sin(theta)/u - Rg*cos(theta))(1 + e^(-2*sqrt(Rg*sin(theta)/u - Rg*cos(theta))*ut/R)))/(1 - e^(-2*sqrt(Rg*sin(theta)/u - Rg*cos(theta))*ut/R)))

now,
assume Rg*sin(theta)/u - Rg*cos(theta)> 0
sin(theta)/u > cos(theta)
tan(theta) > u

hecne as the bead is slipping, theta has to be more than angle of repose r
hence
tan(r) = u
so,
tan(theta) > tan(r)
tahta > r
which is true
hence
Rg*sin(theta)/u - Rg*cos(theta) > 0
henc e
as t -> inf
v = sqrt(Rg*sin(theta)/u - Rg*cos(theta))

now, tan(theta) = h/2*pi*R
sin(theta) = h/sqrt(h^2 + 4*pi^2*R^2)
cos(theta) = 2*pi*R/sqrt(h^2 + 4*pi^2*R^2)
hence
v lim = sqrt(Rg)sqrt(h/u - 2*pi*R)/sqrt(sqrt(h^2 + 4*pi^2*R^2))

so the physicsal reason there should be a terminal velcoity is beause of the rotational omtion of the beadm the normal force on trhe bead is dependent on centripital acceleration which is depenedent of velocity
hence
when normal foce should not be a function of speed, it actually is a function of speed and increasex with increase of speed
hence
this increasing frcition will make the ratre of increase of speed of the bead slower abd hence it shall have a terminal velocity